The Large Time Behivors Of The Solutions To Two Nonlinear Diffusion Equations (Systems)

Diffusion is the spread of substances by the natural movement of their particals, it appears widely in nature. In applied science, many problems related to diffusion can be modelled by partial differential equations with diffusion terms. For instance, these equations arises in many fields such as filtration, phase transition, biochemistry and dynamics of biological groups. Obviously, it is very important to analysis these diffusion equations in views of mathematical theory and applied sciences. In the last four decades, especially in recent twenty years, the study in this direction attracts a large number of mathematician both at home and abroad and develop a series of new mathematical ideas and methods, which enrich enormously the theory of partial differential equations. In this thesis, we will give some qualitative analysis for two degenerate (singular) diffusion equations(systems) arose in applied sciences. The thesis consists of three chapters:The first chapter is devoted to the summary of the dissertation, which recounts the development and the present circumstance about the related problems. Moreover, main results of the present paper are stated.In Chapter 2, we discuss the nonnegative solutions of degenerate parabolic system with nonlinear coupled boundary conditions and nonnegative nontrivial compactly supported initial data, the critical Fujita exponents are given, and furthermore, the blow-up rates of the nonglobal solution are obtained by using Scaling methods.In Chapter 3, we consider the dead-core problem for the fast-diffusion equation with strong absorption u_t=(u^m)_xx-u^P with 0＜p＜m＜1 and positive boundary values. We investigate the dead-core rate, at which the solution reaches its first zero and find that the dead-core rate is faster than the one given by the corresponding ODE. This is contrary to the known results for the related extinction, quenching and blow-up problems. Moreover, we find the dead-core rates quite unstable: the ODE rate can be recovered if the absorption term is replaced by -a(t, x) up for a suitable bounded, uniformly positive function a(t, x). As an application of the above results, we provide some new and relatively simple examples of fast blow-up, and exhibit a phenomenon of strong sensitivity to gradient perturbations. Furthermore, the blow-up rate is found to depend on a constant in the perturbation term, and sharp estimates are also obtained for the profile of dead-core and blow-up.